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Tie Knots as Random Walks

We represent knot sequences as random walks on a triangular lattice. The axes r,c,l correspond to the three regions R,C,L through which the active end can be wound and the unit vectors tex2html_wrap_inline922 represent the corresponding moves (Fig. 3); we omit the directional notation tex2html_wrap_inline924 and the terminal action T. Since all knot sequences end with tex2html_wrap_inline858 and alternate between tex2html_wrap_inline834 and tex2html_wrap_inline836 , all knots of odd half-winding number h begin with tex2html_wrap_inline772 while those of even h begin with tex2html_wrap_inline774 . Our simplified random walk notation is thus unique, and we only make use of the directional notation tex2html_wrap_inline924 in the context of move sequences.

The three-fold symmetry of the move regions implies that only steps along the positive lattice axes are acceptable and, as in the case of moves, no consecutive steps can be identical; the latter condition makes our walk a second order Markov, or persistent, random walk. Nonetheless, every site on the lattice can be reached since, e.g., tex2html_wrap_inline944 and tex2html_wrap_inline946 .

Figure 3: A tie knot may be represented by a persistent random walk on a triangular lattice, beginning with tex2html_wrap_inline948 and ending with tex2html_wrap_inline950 or tex2html_wrap_inline952 . Only steps along the positive r,c and l axes are permitted and no two consecutive steps may be the same. Shown here is the Four-in-Hand, indicated by the walk tex2html_wrap_inline958 .

The evolution equations for the persistent random walk appear as


where tex2html_wrap_inline960 is the conditional probability that the walker is at point (r,c,l) on the nth step, having just taken a step along the positive r-axis, p is conditioned on a step along the positive c-axis, etc. The unconditional probability of occupation of a site, U, may be expressed tex2html_wrap_inline974 .

Yong Mao
Sat Nov 7 16:03:57 GMT 1998